Off-frame animation cycle?
Yes, this is possible.
Create the Action you want to repeat in the Dopesheet.



For clarity, my action is going to be 9.1 frames long and going to repeated 10 times.
Note, that the the length of 9.1 will come from the start and and frames: 1 to 10.1.

endframe - startframe = duration
10.1     -         1  = 9.1



Go to the NLA Editor. 
Press the Push Action Down button. 
Go to the properties panel in the NLA editor and to the Action Clip section.
Set the start frame to your start frame (1)
Set the end frame to your end frame (10.1)
Set the repeat property to the numer of times, you want to repeat the action clip.




The Action will no be repeat with an offset of 0.1 frames.